Question

A pile of gravel is in the shape of a cone with a base diameter of 10 feet and a height of 4 feet.

If 1 cubic foot of gravel weighs 30 pounds, how much does the pile of gravel weigh? Round the answer to the nearest pound.
3,142 pounds
9,425 pounds
12,566 pounds
15,708 pounds

Answer 1

To find the weight of the pile of gravel, we first need to calculate its volume using the formula for the volume of a cone.

The formula for the volume \( V \) of a cone is given by:

\[
V = \frac{1}{3} \pi r^2 h
\]

where \( r \) is the radius of the base and \( h \) is the height of the cone.

1. **Find the radius**: The base diameter is 10 feet, so the radius \( r \) is:

\[
r = \frac{10}{2} = 5 \text{ feet}
\]

2. **Height**: The height \( h \) is given as 4 feet.

3. **Substitute values into the volume formula**:

\[
V = \frac{1}{3} \pi (5^2) (4)
\]

Calculating \( 5^2 \):

\[
5^2 = 25
\]

Now substitute back into the volume formula:

\[
V = \frac{1}{3} \pi (25) (4) = \frac{100}{3} \pi
\]

4. **Approximate the value of \(\pi\)**, using \( \pi \approx 3.14 \):

\[
V \approx \frac{100}{3} \times 3.14 \approx \frac{314}{3} \approx 104.67 \text{ cubic feet}
\]

5. **Weight of gravel**: Since 1 cubic foot of gravel weighs 30 pounds, the total weight \( W \) of the gravel is given by:

\[
W = V \times 30
\]

Substituting the volume we found:

\[
W \approx 104.67 \times 30 \approx 3140.1 \text{ pounds}
\]

Rounding this to the nearest pound gives approximately:

\[
W \approx 3140 \text{ pounds}
\]

However, let’s confirm our calculations by directly multiplying:

\[
104.67 \times 30 = 3140.1 \text{ pounds}
\]

Thus, the final weight of the pile of gravel, rounded to the nearest pound, is approximately:

\[
\text{Weight of the pile of gravel} = \boxed{3140} \text{ pounds}
\]

It appears that none of the provided options (3,142 pounds, 9,425 pounds, 12,566 pounds, 15,708 pounds) match this calculated value. Please ensure the question and options provided are accurate.